PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An adjoint triple induces adjunctions between its associated endofunctors

Statement

If $L ⊣ M ⊣ R$ , with $L, R : D \to C$ and $M : C \to D$ , then

$L M ⊣ R M on C,$

and

$M L ⊣ M R on D .$

These adjunctions use the composite units and counits and require no local-smallness hypothesis.

Facts & Assumptions

Given: An adjoint triple $L ⊣ M ⊣ R$ .

[L1]

An adjoint triple supplies adjunctions $L ⊣ M$ and $M ⊣ R$ with the displayed functor types (Adjoint triple $L ⊣ M ⊣ R$ ).

[L2]

If $F ⊣ G$ and $F^{'} ⊣ G^{'}$ , then $F^{'} F ⊣ G G^{'}$ with the composite unit and counit formulas (Adjunctions compose with the composite unit and counit formulas).

Proof

technique · direct

1.1

Compose $M ⊣ R$ with $L ⊣ M$ in [L2]. The left composite is $L M : C \to C$ and the right composite is $R M : C \to C$ , so $L M ⊣ R M$ .

L1L2

1.2

Compose $L ⊣ M$ with $M ⊣ R$ in [L2]. The left composite is $M L : D \to D$ and the right composite is $M R : D \to D$ , so $M L ⊣ M R$ .

L1L2

2.1

In each case the unit is obtained by inserting the second unit between the first unit's functors, and the counit by inserting the first counit between the second counit's functors, exactly as in [L2]. Since [L2] uses unit-counit data, neither construction requires hom-sets.

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Exercise 4.1.iv (standard reference, not scraped)